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On semisimple quasitriangular Hopf algebras of dimension $dq^n
Abstract
Let be a prime number, d be an odd square-free natural number, and n be a non-negative integer. We prove that a semisimple quasitriangular Hopf algebra of dimension is solvable in the sense of Etingof, Nikshych and Ostrik. In particular, if then it is either isomorphic to for some abelian group G, or twist equivalent to a Hopf algebra which fits into a cocentral abelian exact sequence.
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Let C be a modular category of Frobenius-Perron dimension dqⁿ, where q > 2 is a prime number and d is a square-free integer. We show that C must be integral and nilpotent and therefore group-theoretical. In the case where q = 2, we describe the structure of C in terms of equivariantizations of group-crossed braided fusion categories.
We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of
tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences
of (finite) groups. We classify exact sequences of tensor categories (such that is finite) in terms of normal, faithful Hopf monads on and also in terms of self-trivializing commutative algebras in the center of . More generally, we show that, given any dominant tensor functor admitting an exact (right or left) adjoint, there exists a canonical commutative algebra (A,σ) in the center of such that F is tensor equivalent to the free module functor , where denotes the category of A-modules in endowed with a monoidal structure defined using σ. We re-interpret equivariantization under a finite group action on a tensor category and, in particular, the modularization
construction, in terms of exact sequences, Hopf monads, and commutative central algebras. As an application, we prove that
a braided fusion category whose dimension is odd and square-free is equivalent, as a fusion category, to the representation
category of a group.
This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them pre-modular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras. Comment: 102 pages, LaTeX, typos corrected
We describe all fusion subcategories of the representation category
of a twisted quantum double , where G is a finite group and is a 3-cocycle on G. In view of the fact that every group-theoretical braided fusion category can be embedded into some , this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas
for some invariants of the fusion subcategories of . We also give a characterization of group-theoretical braided fusion categories as equivariantizations of pointed categories.
Using a variety of methods developed in the literature (in particular, the
theory of weak Hopf algebras), we prove a number of general results about
fusion categories in characteristic zero. We show that the global dimension of
a fusion category is always positive, and that the S-matrix of any modular
category (not necessarily hermitian) is unitary. We also show that the category
of module functors between two module categories over a fusion category is
semisimple, and that fusion categories and tensor functors between them are
undeformable (generalized Ocneanu rigidity). In particular the number of such
categories (functors) realizing a given fusion datum is finite. Finally, we
develop the theory of Frobenius-Perron dimensions in an arbitrary fusion
category and classify categories of prime dimension.
We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension (Formula presented.), where (Formula presented.) is a prime number, (Formula presented.) is a square-free natural number and (Formula presented.). We prove that, if (Formula presented.) or (Formula presented.) is prime with (Formula presented.), then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than 1125 is group-theoretical.
We study several classes of braided fusion categories, and prove that they all contain nontrivial Tannakian subcategories. As applications, we classify some fusion categories in terms of solvability and group-theoreticality.
We review some recent results and constructions concerning fusion categories, focusing on the classification problem of semisimple Hopf algebras. We discuss several invariants of these structures, as well as some open questions.
We show that the Witt class of a weakly group-theoretical non-degenerate
braided fusion category belongs to the subgroup generated by classes of
non-degenerate pointed braided fusion categories and Ising braided categories.
This applies in particular to solvable non-degenerate braided fusion
categories. We also give some sufficient conditions for a braided fusion
category to be weakly group-theoretical or solvable in terms of the
factorization of its Frobenius-Perron dimension and the Frobenius-Perron
dimensions of its simple objects. As an application, we prove that every
non-degenerate braided fusion category whose Frobenius-Perron dimension is a
natural number less than 1800, or an odd natural number less than 33075, is
weakly group-theoretical.
Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq
r
. We conclude the classification of semisimple Hopf algebras A of dimension pq
2 over an algebraically closed field k of characteristic zero, such that both A and A
* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq
2<100.
We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups – weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group-theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq2, where p,q,r are distinct primes.
In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger.
Incluye bibliografía e índice
Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use this statement to prove that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily the result of Zhu that Kaplansky's conjecture on prime dimensional Hopf algebras over k is true, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne on characterization of tannakian categories to prove that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.
We study a natural construction of Hopf algebra quotients canonically associated to an R-matrix in a finite-dimensional Hopf algebra. We apply this construction to show that a quasitriangular Hopf algebra whose
dimension is odd and square-free is necessarily cocommutative.
We classify the module categories over the double (possibly twisted) of a finite group.
We determine the structure of Hopf algebra extensions of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs. Comment: Main changes appear in the description in Theorem 1.1, missing case added in proof of Proposition 6.1 and reference added. To appear in Algebr. Represent. Theory
Removed because of inappropriateness for e-print archives.
- C Kassel
C. Kassel, Quantum groups, GTM 155 (1995).